Do any antibiotics make you feel dizzy like you are walking on a boat?

Even on dry land. Most medicines list dizziness as a potential side-effect but some antibiotics such as Minocycline have a greater possibility of causing dizziness. Higher concentrations of the drug affect the balancing system of the inner ear which cause the abnormal sensation.
Yes. The classic example is minocycline, which can cause vertigo in about 8 -10% of patients taking it. It is related in part to the serum levels which penetrate into the vestibular apparatus. Taking lower doses four times a day rather than a high dose twice a day seems to minimize this. Long term treatment or high doses of aminoglycosides can produce permanent damage. There are others.